Step |
Hyp |
Ref |
Expression |
1 |
|
ldilcnv.h |
|- H = ( LHyp ` K ) |
2 |
|
ldilcnv.d |
|- D = ( ( LDil ` K ) ` W ) |
3 |
|
simpll |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> K e. HL ) |
4 |
|
eqid |
|- ( LAut ` K ) = ( LAut ` K ) |
5 |
1 4 2
|
ldillaut |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> F e. ( LAut ` K ) ) |
6 |
4
|
lautcnv |
|- ( ( K e. HL /\ F e. ( LAut ` K ) ) -> `' F e. ( LAut ` K ) ) |
7 |
3 5 6
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> `' F e. ( LAut ` K ) ) |
8 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
9 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
10 |
8 9 1 2
|
ldilval |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) -> ( F ` x ) = x ) |
11 |
10
|
3expa |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) -> ( F ` x ) = x ) |
12 |
11
|
3impb |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( F ` x ) = x ) |
13 |
12
|
fveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( `' F ` ( F ` x ) ) = ( `' F ` x ) ) |
14 |
8 1 2
|
ldil1o |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) ) |
15 |
14
|
3ad2ant1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) ) |
16 |
|
simp2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> x e. ( Base ` K ) ) |
17 |
|
f1ocnvfv1 |
|- ( ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ x e. ( Base ` K ) ) -> ( `' F ` ( F ` x ) ) = x ) |
18 |
15 16 17
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( `' F ` ( F ` x ) ) = x ) |
19 |
13 18
|
eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( `' F ` x ) = x ) |
20 |
19
|
3exp |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> ( x e. ( Base ` K ) -> ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) ) |
21 |
20
|
ralrimiv |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> A. x e. ( Base ` K ) ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) |
22 |
8 9 1 4 2
|
isldil |
|- ( ( K e. HL /\ W e. H ) -> ( `' F e. D <-> ( `' F e. ( LAut ` K ) /\ A. x e. ( Base ` K ) ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) ) ) |
23 |
22
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> ( `' F e. D <-> ( `' F e. ( LAut ` K ) /\ A. x e. ( Base ` K ) ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) ) ) |
24 |
7 21 23
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> `' F e. D ) |